Local non-Gaussianities in the initial conditions of the Universe, parameterized by $f_{\rm NL}$, induce a scale-dependence in the large-scale bias of halos in the late Universe. This effect is a promising path to constrain multi-field inflation theories that predict non-zero $f_{\rm NL}$. While most existing constraints from the halo bias involve auto-correlations of the galaxy distribution, cross-correlations with probes of the matter density provide an alternative channel with fewer systematics. We present the strongest large-scale structure constraint on local primordial non-Gaussianity that uses cross-correlations alone. We use the cosmic infrared background (CIB) consisting of dusty galaxies as a halo tracer and cosmic microwave background (CMB) lensing as a probe of the underlying matter distribution, both from \textit{Planck} data. Milky Way dust is a key challenge in using the large-scale modes of the CIB. Importantly, the cross-correlation of the CIB with CMB lensing is far less affected by Galactic dust compared to the CIB auto-spectrum, which picks up an additive bias from Galactic dust. We find no evidence for primordial non-Gaussianity and find $-87Comment: 16 pages, 9 figures